Regular representations of completely bounded maps
We study properties and the structure of some special classes of homomorphisms on C*-algebras. These maps are *-preserving up to conjugation by a symmetry. Making use of these homomorphisms, we prove a new structure theorem for completely bounded maps from a unital C*-algebra into the algebra of all bounded linear maps on a Hilbert space. Finally we provide alternative proofs for some of the known results about completely bounded maps and improve on them.
C∗-extreme points of entanglement breaking maps
In this paper, we study the C∗-convex set of unital entanglement breaking (EB-)maps on matrix algebras. General properties and an abstract characterization of C∗-extreme points are discussed. By establishing a Radon-Nikodym-type theorem for a class of EB-maps we give a complete description of the C∗-extreme points. It is shown that a unital EB-map: Md1 -Md2 is C∗-extreme if and only if it has Choi-rank equal to d2. Finally, as a direct consequence of the Holevo form of EB-maps, we derive a non-commutative analog of the Krein-Milman theorem for C∗-convexity of the set of unital EB-maps.
Mapping cone of k-entanglement breaking maps
In Christandl et al. (Ann Henri Poincaré 20(7):2295–2322, 2019), the authors introduced k-entanglement breaking linear maps to understand the entanglement breaking property of completely positive maps on taking composition. In this article, we do a systematic study of k-entanglement breaking maps. We prove many equivalent conditions for a k-positive linear map to be k-entanglement breaking, thereby study the mapping cone structure of k-entanglement breaking maps. We discuss examples of k-entanglement breaking maps and some of their significance. As an application of our study, we characterize the completely positive maps that reduce Schmidt number on taking composition with another completely positive map. Finally, we extend a spectral majorization result for separable states.
On a generalization of Ando's dilation theorem
We introduce the notion of Q-commuting operators which includes commuting operators. We prove a generalized version of the commutant lifting theorem and Ando's dilation theorem in the context of Q-commuting operators.